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FRM - Schweser - Topic 41
The Black-Scholes Merton model
6
Finance
Professional
05/05/2010
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Term
What does BSM model assume?
 Definition
stock prices are lognormally distributed. Stock returns are normally distributed.
Term
Assume a stock has an expected return of 12% and an annual std dev of 20%. Calculate the mean and std dev of the probability distribution over a 4 year period:
 Definition

continuously compounded annual returns have a mean of:

 

(μ - (σ2/2))

 

= .12 - (.22/2)

= .1 = 10%

 

Std deviation = σ / √T

= .2 / sqrt(4)

= .2/2

= .10 = 10%
Term
Assume a stock is currently priced at $25 with an expected annual return of 20%. Calculate the expected value of the stock in 6 months:
 Definition

E(ST) = S0eμT

 

= $25e.2*.5

 

= $27.63

 

Note - the mean return will be always slightly less than the expected return
Term
Consider a portfolio that has the following asset returns: 5%, -4%, 9%, 6%. Calculate the return realised by this portfolio:
 Definition

use a geometric mean

 

= (1.05 * .96 * 1.09 * 1.06).25 - 1

= 3.9%
Term
What are the assumptions of the black scholes merton option pricing model?
 Definition

- the price of the asset follows a lognormal distribution

- the (continuous) risk free rate is constant and known

- the volatility of the underlying asset is constant and known

- markets are frictionless

- the underlying asset has no cash flow

the options are european options  (BSM model does not correctly price american options)
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